A Structured Learning Approach to Attributed Graph Embedding

  • Haifeng Zhao
  • Jun Zhou
  • Antonio Robles-Kelly
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6218)


In this paper, we describe the use of concepts from structural and statistical pattern recognition for recovering a mapping which can be viewed as an operator on the graph attribute-set. This mapping can be used to embed graphs into spaces where tasks such as categorisation and relational matching can be effected. We depart from concepts in graph theory to introduce mappings as operators over graph spaces. This treatment leads to the recovery of a mapping based upon the graph attributes which is related to the edge-space of the graphs under study. As a result, this mapping is a linear operator over the attribute set which is associated with the graph topology. Here, we employ an optimisation approach whose cost function is related to the target function used in discrete Markov Random Field approaches. Thus, the proposed method provides a link between concepts in graph theory, statistical inference and linear operators. We illustrate the utility of the recovered embedding for shape matching and categorisation on MPEG7 CE-Shape-1 dataset. We also compare our results to those yielded by alternatives.


Cost Function Target Space Markov Random Field Graph Topology Graph Match 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)CrossRefGoogle Scholar
  2. 2.
    Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)CrossRefGoogle Scholar
  3. 3.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: NIPS. Number, vol. 14, pp. 634–640 (2002)Google Scholar
  4. 4.
    Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  5. 5.
    Sebastian, T.B., Klein, P.N., Kimia, B.B.: Shock-based indexing into large shape databases. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2352, pp. 731–746. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  6. 6.
    Wong, A.K.C., You, M.: Entropy and distance of random graphs with application to structural pattern recognition. IEEE TPAMI 7, 599–609 (1985)zbMATHGoogle Scholar
  7. 7.
    Christmas, W.J., Kittler, J., Petrou, M.: Structural matching in computer vision using probabilistic relaxation. IEEE TPAMI 17(8), 749–764 (1995)Google Scholar
  8. 8.
    Wilson, R., Hancock, E.R.: Structural matching by discrete relaxation. IEEE TPAMI 19(6), 634–648 (1997)Google Scholar
  9. 9.
    Caetano, T., Cheng, L., Le, Q., Smola, A.: Learning graph matching. In: ICCV, pp. 14–21 (2007)Google Scholar
  10. 10.
    Biggs, N.L.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1993)Google Scholar
  11. 11.
    Bremaud, P.: Markov Chains, Gibbs Fields, Monte Carlo Simulation and Queues. Springer, Heidelberg (2001)Google Scholar
  12. 12.
    Keuchel, J.: Multiclass image labeling with semidefinite programming. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3952, pp. 454–467. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Kumar, M., Torr, P., Zisserman, A.: Solving markov random fields using second order cone programming relaxations. In: CVPR, pp. 1045–1052 (2006)Google Scholar
  14. 14.
    Cour, T., Shi, J.: Solving markov random fields with spectral relaxation. In: Intl. Conf. on Artificial Intelligence and Statistics (2007)Google Scholar
  15. 15.
    Young, G., Householder, A.S.: Discussion of a set of points in terms of their mutual distances. Psychometrika 3, 19–22 (1938)CrossRefGoogle Scholar
  16. 16.
    Ding, C., He, X.: K-means clustering via principal component analysis. In: ICML, pp. 225–232 (2004)Google Scholar
  17. 17.
    Gold, S., Rangarajan, A.: A graduated assignment algorithm for graph matching. IEEE TPAMI 18(4), 377–388 (1996)Google Scholar
  18. 18.
    Demirci, M.F., Shokoufandeh, A., Dickinson, S.J.: Skeletal shape abstraction from examples. IEEE TPAMI 31(5), 944–952 (2009)Google Scholar
  19. 19.
    Belongie, S., Malik, J., Puzicha, J.: Shape matching and object recognition using shape contexts. IEEE TPAMI 24(24), 509–522 (2002)Google Scholar
  20. 20.
    Chen, L., McAuley, J.J., Feris, R.S., Caetano, T.S., Turk, M.: Shape classification through structured learning of matching measures. In: CVPR (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Haifeng Zhao
    • 1
  • Jun Zhou
    • 2
    • 3
  • Antonio Robles-Kelly
    • 2
    • 3
  1. 1.School of Comp. Sci. & Tech.Nanjing Univ. of Sci. & Tech.NanjingChina
  2. 2.Australian National UniversityCanberraAustralia
  3. 3.National ICT Australia (NICTA)CanberraAustralia

Personalised recommendations